Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 1 / i / Solution

Take a compactly supported smooth variation $\phi \in C_c^{\infty }(\Omega )$ and set $u_t=u+t\phi$. Its support is away from the boundary, so it preserves any prescribed boundary values. Differentiating the functional at $t=0$ gives the first variationBy integration by parts, there is no boundary term. The fundamental lemma of the calculus of variations therefore gives the Euler-Lagrange equationExpanding the total derivative, an equivalent classical form isRepeated indices are summed. In the last term involving $x_i$, the derivative holds $z,p$ fixed; the other terms account for their dependence on $u(x),Du(x)$. Compactly supported variations derive the interior equation without imposing a natural boundary condition.